Polar analog-to-digital converter and down converter for bandpass signals

ABSTRACT

Methods and systems for generating a digital representation of the amplitude and phase of a bandpass signal are disclosed. The methods comprise filtering the bandpass signal with a bandpass filter, generating the real and imaginary parts of the complex analytic signal with a quadrature hybrid, determining the amplitude of the complex analytic signal by adding an even power-law transform of the real and imaginary parts of the complex analytic signal, and determining the phase of the complex analytic signal by comparing the real and imaginary parts of the complex analytic signal to zero and comparing an even power-law transform of the real and imaginary parts of the complex analytic signal to each other. Analog to digital converters and methods of converting complex analytic signals to digital signals are also disclosed.

REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. Non-Provisional applicationSer. No. 16/218,845, filed Dec. 13, 2018, which is a continuation ofU.S. Non-Provisional application Ser. No. 15/919,962, filed Mar. 13,2018, which claims priority to U.S. Provisional Application No.62/471,585, filed Mar. 15, 2017, all entitled “Polar Analog-To-DigitalConverter and Down Converter for Bandpass Signals,” and herebyspecifically and entirely incorporated by reference.

RIGHTS IN THE INVENTION

This invention was made with government support under CooperativeAgreement AST-1519126, between the National Science Foundation andAssociated Universities, Inc., and, accordingly, the United Statesgovernment has certain rights in this invention.

BACKGROUND 1. Field of the Invention

This invention is directed to analog to digital converters (ADC) anddown converters for bandpass signals. More specifically, the inventionis directed to generating the analog Hilbert transform of the bandpasssignal by means of a quadrature (90-degree) hybrid.

2. Description of the Background

Sampling broadband signals with high relatively bandwidths is acomplicated task. One approach is to consider the signal as a basebandsignal and sample the signal at least at the Nyquist sampling rate. Fora relative bandwidth of 100%, this implies a sampling rate at least 3times the signal bandwidth. A second approach, known as undersampling orbandpass sampling, consists of directly sampling the RF/IF signal usinga sampling frequency such that the frequency band of interest isentirely contained within the limits of one “Nyquist zone”. In general,this requires a higher sampling frequency than the method proposed andis possible (with no additional hardware) only for a small enoughrelative bandwidth. A third approach would be to downconvert the signal,but once again, additional hardware is required and a good linearity canbe achieved only if the relative bandwidth is small.

In some existing methods to sample broadband signals a Local Oscillator(LO) signal is inputted into a 90-degree hybrid whose outputs are thenmixed with the input RF/IF signal, e.g. see (FIG. 4 of U.S. Pat. No.7,356,103).

Another approach is to digitize directly the bandpass signal as if itwere a baseband signal, using a sampling rate faster than the highestfrequency. No quadrature hybrid is used in this method. For example,U.S. Pat. No. 4,982,193 provides a similar method to digitize at thecarrier frequency, or preferably, one of its harmonics. Also, U.S. Pat.Nos. 8,279,100 and 7,692,570 each provide systems to directly digitizean RF signal at sampling rates commensurable with the signal bandwidthby means of a plurality of ADCs.

As another example of the existing methods, U.S. Pat. No. 6,466,150provides an ADC that is suited for digitizing the phase of the signal.The method exclusively focuses on digitizing the phase only, withapplication to PM or FM signals. Moreover, the digitizer operates onbaseband (downconverted) signals and does not utilize a full-waverectifier, leading to more comparators (more power consumption) than theproposed method.

US Patent Application No. 2013/0069812 discloses an alternative methodto digitize bandpass signals located in any Nyquist zone by means of atime-interleaved analog-to-digital converter. It applies totime-interleaved ADC is used and the Hilbert transform is applied in thedigital domain, so the analog input is not a complex signal. This methodsuffers from the need of a calibration method inherent to itstime-interleaved nature.

U.S. Pat. No. 8,482,445 also applies to time-interleaved ADCs, and henceit is also hindered by the need of complex calibration methods. Thecalibration method disclosed is a time-interleaved ADC by the injectionof a probe tone at a different Nyquist zone. The Hilbert transform isapplied in the digital domain.

U.S. Pat. No. 6,301,312 is directed to phase-locked loops. It employs aconventional IQ demodulation where the IF signal is mixed with thein-phase and quadrature versions of a local oscillator. This system usesmixers which can lead to mixer insertion loss, typically about 20 dB.

SUMMARY

The present invention overcomes the problems and disadvantagesassociated with current strategies and designs and provides new toolsand methods for analog to digital converters (ADC) and down convertersfor bandpass signals.

One embodiment of the invention is directed to a method of determiningan amplitude and phase of a complex analytic signal. The methodcomprises the steps of filtering the complex analytic signal with abandpass filter, generating the real and imaginary parts of the complexanalytic signal with a quadrature hybrid, determining the amplitude ofthe complex analytic signal by adding the real and imaginary parts ofthe complex analytic signal, and determining the phase of the complexanalytic signal by comparing the real and imaginary parts of the complexanalytic signal to zero and comparing an even power-law transform of thereal and imaginary parts of the complex analytic signal to each other.

In a preferred embodiment, an amplitude generator circuit extracts thesignal amplitude or a power-law thereof. Preferably, the amplitudegenerator is comprised of a lowpass filter to attenuate generatedharmonics and a power-law inverter. The method preferably furthercomprises converting the amplitude of the complex analytical signal intobinary signals with a linear quantizer. Preferably, the amplitudegenerator is comprised of a nonlinear quantizer.

In a preferred embodiment, the step of comparing an even power-lawtransform of the real and imaginary parts of the complex analytic signalto each other further comprises multiplying the even power-law transformof the real and imaginary parts of the complex analytic signal bymultiple constant gains in their respective gain banks. The methodpreferably further comprises comparing scaled versions of the real andimaginary parts of the complex analytic signal with a bank ofcomparators to generate a phase binary signal.

Another embodiment of the invention is directed to a system adapted tooutput an amplitude and phase of a complex analytic signal. The systemcomprises a bandpass filter adapted to filter the complex analyticsignal, a quadrature hybrid adapted to generate the real and imaginaryparts of the complex analytic signal, a power-law device adapted todetermine the amplitude of the complex analytic signal by adding thereal and imaginary parts of the complex analytic signal, and a bank ofcomparators adapted to determine the phase of the complex analyticsignal by comparing the real and imaginary parts of the complex analyticsignal to zero and comparing an even power-law transform of the real andimaginary parts of the complex analytic signal to each other.

Preferably the system further comprises an amplitude generator circuitadapted to extract the signal amplitude or a power-law thereof. In apreferred embodiment, the amplitude generator is comprised of a lowpassfilter to attenuate generated harmonics and a power-law inverter. Thesystem preferably further comprises a linear quantizer adapted toconvert the amplitude of the complex analytic signal into binarysignals. The amplitude generator is preferably comprised of a nonlinearquantizer.

In a preferred embodiment the system comprises comparing an evenpower-law transform of the real and imaginary parts of the complexanalytic signal to each other further comprises multiplying the evenpower-law transform of the real and imaginary parts of the complexanalytic signal by multiple constant gains in their respective gainbanks. The system preferably further comprises a bank of comparators togenerate a phase binary signal.

Another embodiment of the invention is directed to a method ofconverting a bandpass analog signal to a digital signal. The methodcomprises the steps of generating the real and imaginary parts of acomplex analytic signal with a quadrature hybrid, determining anamplitude of the complex analytic signal by adding an even power-lawtransform of the real and imaginary parts of the complex analyticsignal, converting the signal amplitude into a digital amplitude signalwith a quantizer, and converting a signal phase of the complex analyticsignal into a digital phase signal by comparing the real and imaginaryparts of the complex analytic signal to zero and comparing an evenpower-law transform of the real and imaginary parts of the complexanalytic signal to each other.

Preferably, the amplitude is lowpass filtered to attenuate generatedharmonics previous to the quantizer. In a preferred embodiment, apower-law inverter transforms the amplitude previous to the quantizer.The method preferably further comprises converting the amplitude of thecomplex analytic signal into binary signals with a linear quantizer. Ina preferred embodiment, the amplitude generator is comprised of anonlinear quantizer.

Preferably, the step of comparing an even power-law transform of thereal and imaginary parts of the complex analytical signal to each otherfurther comprises multiplying the even power-law transform of the realand imaginary parts of the complex analytical signal by multipleconstant gains in their respective gain banks. The method preferablyfurther comprises comparing scaled versions of an even power-lawtransform of the real and imaginary parts of the complex analytic signalwith a bank of comparators to generate a digital phase signal withimproved resolution. Preferably, at least one track-and-hold circuitholds a signal as dictated by a clock signal. Preferably the methodfurther comprises preventing transitions from occurring within a clockcycle with a synchronizer.

Another embodiment of the invention is directed to an analog to digitalconverter (ADC), acting on a bandpass signal and its Hilbert transform.The ADC comprises a first power-law device adapted to apply an evennon-linear transform on the input bandpass signal, a second power-lawdevice adapted to apply an even non-linear transform on the inputHilbert transform signal, an amplitude generator adapted to determine asignal amplitude of the complex analytic signal of the input signal byadding the outputs of said power-law devices, a quantizer adapted toconvert the output of said amplitude generator into a digital amplitudesignal, an encoder adapted to reduce the number of bit signals used torepresent the digital amplitude signal, and a bank of comparatorsadapted to generate a digital signal phase of the complex analyticsignal by comparing the both input bandpass signal and its Hilberttransform to zero, and comparing the outputs of said power-law devicesto each other.

Preferably, the amplitude generator is further comprised of a lowpassfilter to attenuate generated harmonics. In a preferred embodiment, theamplitude generator is further comprised of a power-law inverter. TheADC preferably further comprises a linear quantizer adapted to convertthe amplitude of the complex analytical signal into binary signals.Preferably, the amplitude generator is comprised of a nonlinearquantizer.

In a preferred embodiment the ADC comparing the outputs of the power-lawdevices further comprises multiplying the outputs by multiple constantgains in their respective gain banks, and the ADC further comprises: abank of comparators to generate a digital phase signal and an encoderadapted to reduce the number of bit signals used to represent saiddigital phase signal. Preferably, the ADC further comprises at least onetrack-and-hold circuit adapted to hold a signal as dictated by a clocksignal. Preferably, the ADC further comprises a synchronizer adapted toprevent transitions from occurring within a clock cycle with.

Other embodiments and advantages of the invention are set forth in partin the description, which follows, and in part, may be obvious from thisdescription, or may be learned from the practice of the invention.

DESCRIPTION OF THE FIGURES

FIG. 1 depicts an embodiment of a complex sampling and demodulationusing a quadrature hybrid.

FIG. 2 depicts an embodiment of a polar analog-to-digital conversion anddown conversion using a quadrature hybrid and an amplitude and phaseADC.

FIG. 3 depicts an embodiment of a graphical representation of the phasequantization method.

FIG. 4 depicts an embodiment of a functional diagram of system describedherein.

FIG. 5 depicts an embodiment of an analog-to-digital converter based onthe invention.

FIG. 6 depicts an embodiment of an implementation of the invention.

DESCRIPTION OF THE INVENTION

The systems and methods disclosed aim to solve the problem ofanalog-to-digital conversion of a bandpass signal with minimum samplingfrequency. The method includes generating the analog Hilbert transformof the bandpass signal by means of a quadrature (90 degree) hybrid. Thetwo outputs of the quadrature hybrid (the bandpass signal and itsHilbert transform) can be thought of as the real and imaginary parts ofa complex signal. When such a complex signal is sampled at a rate equalto the signal bandwidth, a baseband copy of the signal willautomatically be generated through aliasing.

The method consists of sampling the magnitude and phase of the complexsignal, which are generated from the hybrid's outputs (the real andimaginary parts). The main advantage of sampling magnitude and phase (ascompared to sampling real and imaginary parts) is that both samplers canshare part of their circuitry, and hence decreases the total powerconsumption. Moreover, the precision of the converter is improved as thephase can be extracted by comparing the real and imaginary parts to eachother, thereby removing the need of comparing the output to a calibratedabsolute value. In addition, other nonidealities of the quantizer, suchas nonlinearity and DC offset, are mitigated in the amplitude and phaseresult. Compared to baseband sampling and undersampling, the methodutilizes a lower sampling rate, thereby avoiding some practicallimitations due to circuitry speed. Finally, the system preferably usesless hardware than an IQ downconverter and sampler yet does not sufferfrom DC bias.

Any real signal has a complex analytic signal associated to it, whichcan be obtained by adding the signal's Hilbert transform for theimaginary part. The Hilbert transform can be obtained through aquadrature (or 90-degree) hybrid, as shown in FIG. 1. Therefore, bysampling the two outputs of the quadrature hybrid at a sampling ratethat is at least the signal bandwidth, a copy of the spectrum is aliasedto baseband for further digital processing. For example, as shown inFIG. 1, an input bandpass signal (as represented by the spectrumsymmetrical around the frequency origin) enters a low noise amplifierand is fed into a bandpass filter. The filtered signal is then fed intoa 90-degree hybrid to obtain the Hilbert transform. The two outputs ofthe quadrature hybrid represent the real and imaginary parts of acomplex signal whose spectrum is the input's spectrum where the negativefrequency components have been cancelled out. Alternatively, thepositive frequency components can be removed (while preserving thenegative ones) by swapping the outputs of the hybrid. The real andimaginary outputs of the 90-hybrid are then fed to two separate ADCs ata sampling rate that is at least the bandwidth of the signal, in orderto alias the signal to baseband for further digital processing. Theresulting aliased spectrum after digitization is shown in FIG. 1 for asampling frequency equal to the input bandpass signal's bandwidth. Inpractice, the negative frequency components (or the positive ones in thealternative configuration) cannot be perfectly removed by the hybrid'saction and perturb the final result. However, this perturbation can beremoved in the digital domain by means of a compensation filter, asexplained, for instance, in [F. E. Churchill et al., IEEE Transactionson Aerospace Electronic Systems, vol. AES-17, no. 1, pp. 131-137,January 1981].

One embodiment of the current invention is directed to generating andsampling the amplitude and phase of the complex analytic signal insteadof its real and imaginary parts. FIG. 2 shows an example of a completesystem where the real and imaginary parts of the complex analytic signalare generated through a quadrature hybrid, as in FIG. 1, and, instead,serve as inputs of the polar analog-to-digital converter and downconverter. A detailed description of said converter follows.

One advantage of the polar analog-to-digital converter is that bothamplitude and phase generation circuits share part of the circuitry andhence decrease both the power consumption and the amplitude-phaseimbalance. The principle of generation is preferably as follows: Asymmetrical (even function) power-law transformation is applied to boththe real and imaginary inputs:

R _(e)(t)=|R(t)|^(α)

I _(e)(t)=|I(t)|^(α)  (1)

where α∈

except zero. The amplitude signal can be preferably generated from thesum of R_(e)(t) and I_(e)(t) as follows:

A _(e)(t)=R _(e)(t)+I _(e)(t)=|R(t)|^(α) +|I(t)|^(α) =A ^(α)(t)[|cosφ(t)|^(α)+|sin φ(t)|^(α)]  (2)

where A(t) and φ(t) are the amplitude and phase, respectively, of theADC's input signal. It can be shown that the term within bracketscontains a constant term in addition to spectral components at multiplesof the fourth harmonic of the signal component. Thus, those unwantedharmonics can be filtered out by means of a lowpass filter or simplyneglected. In the latter case, assuming α is close enough to 2, theharmonic components vanish because the term within brackets in eq. (2)is equal to 1, and hence:

A _(e)(t)=A ²(t)  (3)

In any case, the amplitude signal, A_(e)(t), or a power-lawtransformation thereof can be readily sent to the analog-to-digitalconverter's quantizer.

With regards to the phase signal, comparing both the real and imaginarypart to zero will extract their sign, and hence the information aboutwhich quadrant of the unit circle the phase is located. Additionally,the phase within a quadrant can be determined from comparing the evenpower-law transformation of the real and imaginary parts to each other.For example, from the comparison:

R ^(e)(t)

I _(e)(t)⇔|R(t)|

|I(t)|  (4)

along with the signs of the real and imaginary parts, one can determineif the phase is greater or less than 45, 135, −45, and −135 degrees.Consequently, by comparing the even transformation of the real andimaginary parts instead of the real and imaginary parts themselves, thenumber of required comparisons decreases from 2^(b-1) to (2^(b-2)+1),i.e., approximately half the number of comparisons are required.Therefore, the power consumption can preferably be reduced by half aswell. The rationale behind the phase quantization is representedgraphically in FIG. 3 for a 4-bit quantization example.

FIG. 4 depicts a functional block diagram of an embodiment of a systemutilizing the method described herein. As explained in the foregoingdiscussion, the inputted real and imaginary parts are compared to zeroin order to extract their sign, and then are preferably passed through apower-law device. From the addition of the power-law devices, anamplitude generator circuit preferably extracts the signal amplitude, ora power-law thereof. Such a device is highly dependent on the power, α,and the required spurious level. In general, it is composed of a lowpassfilter to attenuate the generated harmonics, and a power-law inverter,e.g., the square root if α=2. Then a linear quantizer preferablyconverts the analog amplitude into binary signals. Alternatively, thepower-law inverter can be removed, and a nonlinear quantizer is usedinstead. To generate the phase bits, the outputs of the power-lawdevices are preferably multiplied by multiple constant gains in theirrespective gain banks. For example, 1 and (0.414)^(α) are the gainspreferably needed for a 4-bit phase quantization (see FIG. 3). Finally,a bank of comparators preferably compares the scaled versions ofR_(e)(t) and I_(e)(t) to generate the phase binary signals, which jointhe sign signals mentioned before.

By itself, the embodiment sketched in FIG. 4 does not constitute acomplete analog-to-digital converter. FIG. 5 shows a possible embodimentof such an ADC based on the present invention. First, the voltages ofthe real and imaginary parts are preferably held at specific samplinginstants dictated by a clock signal through their respectivetrack-and-hold circuits. Then, the proposed invention, as shown in FIG.4, preferably generates the quantized amplitude and phase binarysignals. A subsequent track-and-hold bank (e.g. a bank of flip-flops ora sample and hold circuit) preferably holds these signals for the entireclock cycle. Thereupon, an encoder preferably converts the results ofthe comparators into meaningful bits, and finally, a synchronizer(flip-flop bank) may be used to prevent transitions from occurringwithin a clock cycle in lieu of at its boundaries. Note that therelative location of the track-and-hold circuitry with respect to thepresent invention can be varied with respect to FIG. 5, possiblyimpacting the maximum speed of the system. For instance, a singlesample-and-hold circuit could be located either at the input or outputof the present invention while replacing both track-and-hold circuits inFIG. 5. Additionally, the track-and-hold circuitry could be integratedwith the different components of the exemplary embodiment in FIG. 4. Theamplitude-and-phase ADC shown in FIG. 5 can be used as previously shownin FIG. 2 to create a polar analog-to-digital converter and demodulatorfor bandpass signals.

FIG. 6 plots one example of a circuit implementing the embodiment fromFIG. 4. Both the real and imaginary inputs are connected to theirrespective differential buffers. The outputs thereof are connected totheir respective sign extractors and power-law devices. In this example,the sign device is a latch comparator and the power-law chosen is α=2.The outputs of the power-law devices are connected to a bank of latchcomparator, a resistor-based gain bank, and an amplitude generatorcircuit. The resistor-based (could be capacitor based as well) gain bankscales the squared input by a factor of (0.414)^(α) by proper choice ofthe relative resistor's values (R2=4.8·R1). Said scaled versions areinputted to the comparator network where a bank of latch comparatorsgenerates the output binary signals (in addition to the sign of theinputs) according to the phase quantization method represented in FIG.3. The outputs of the squaring devices are added by the first invertingamplifier of the amplitude generator circuit. Said inversion is undoneby a second inverting amplifier which serves as input buffer of theamplitude quantizer as well. The output of amplitude generator is thesquared amplitude, and hence the quantizer is nonlinear (quadratic),i.e., it compares the squared amplitude with reference voltages whichproduce a thermometer coding of the amplitude in uniformly spacedquantization intervals. Said reference voltages can be generated bymeans of a reference voltage applied to a set of resistors in serieswith proper relative values (as indicated in FIG. 6). A set of latchcomparators produce the binary signals representing the quantizedamplitude using thermometer coding. For example, the reference voltageat the lowest comparator is V_(DD)×R4/(28 R3+R4)=V_(DD)/255, if R3=8·R4.As a result, the binary output of the first comparator is equal to A²

V_(DD)255, or equivalently, A

√{square root over (V_(DD))}/15. Similarly, the next reference levelbecomes: V_(DD)×(R3+R4)/(28 R3+R4)=V_(DD)×9/255, and its correspondingbinary output, A

3√{square root over (V_(DD))}/15. By defining, V_(REF)

2√{square root over (V_(DD))}/15, the binary signals indicated in FIG. 6are obtained.

Digitizing the amplitude and phase instead of the real and imaginaryparts produces a simplified circuitry over traditional circuits, whichpreferably reduces the power consumption and increases the analogbandwidth of the converter. One polar ADC substitutes two independentADCs for the real and imaginary parts, and thus, avoids problems arisingfrom poor calibration and/or synchronization between said ADCs. Thesystem has general applications as an analog-to-digital converter, andit is suitable for simplifying various subsequent digital signalprocessing, such as digital PLLs, PM/FM demodulators, etc., where phaseextraction or complex multiplication, for instance, are the first stepsin the processing chain.

Other embodiments and uses of the invention will be apparent to thoseskilled in the art from consideration of the specification and practiceof the invention disclosed herein. All references cited herein,including all publications, U.S. and foreign patents and patentapplications, are specifically and entirely incorporated by reference.The term comprising, where ever used, is intended to include the termsconsisting and consisting essentially of. Furthermore, the termscomprising, including, and containing are not intended to be limiting.It is intended that the specification and examples be consideredexemplary only with the true scope and spirit of the invention indicatedby the following claims.

1. A method of determining an amplitude and phase of a bandpass signal, comprising the steps of: generating real and imaginary parts of an analytic representation of said bandpass signal through a quadrature hybrid, wherein the real and imaginary parts of an analytic representation of said bandpass signal is an analytic signal; determining an amplitude of the analytic signal by adding an even power-law transform of the real and imaginary parts of the analytic signal; and determining a phase of the analytic signal by comparing the real and imaginary parts of the analytic signal to zero and comparing the even power-law transform of an absolute value of the real and imaginary parts of the analytic signal to each other.
 2. The method of claim 1, wherein an amplitude generator circuit extracts the signal amplitude or a power-law thereof.
 3. The method of claim 2, wherein the amplitude generator is comprised of a lowpass filter to attenuate generated harmonics and a power-law inverter.
 4. The method of claim 3, further comprising converting the amplitude of the analytic signal into binary signals with a linear quantizer.
 5. The method of claim 2, wherein the amplitude generator is comprised of a nonlinear quantizer.
 6. The method of claim 1, wherein the step of comparing an even power-law transform of the real and imaginary parts of the analytic signal to each other further comprises multiplying the even power-law transform of the real and imaginary parts of the analytic signal by multiple constant gains in their respective gain banks.
 7. The method of claim 6, further comprising comparing scaled versions of the real and imaginary parts of the analytic signal with a bank of comparators to generate a phase binary signal.
 8. The method of claim 1, where the even power-law transform of the real and imaginary parts of the analytic signal are logarithmically converted prior to their comparison to each other.
 9. A system adapted to output an amplitude and phase of a bandpass signal, comprising: a quadrature hybrid adapted to generate real and imaginary parts of an analytic representation of said bandpass signal, wherein the real and imaginary parts of an analytic representation of said bandpass signal is an analytic signal; a power-law device adapted to determine an amplitude of the analytic signal by adding an even power-law transform of the real and imaginary parts of the analytic signal; and a bank of comparators adapted to determine a phase of the analytic signal by comparing the real and imaginary parts of the analytic signal to zero and comparing the even power-law transform of the real and imaginary parts of the analytic signal to each other.
 10. The system of claim 9, further comprising an amplitude generator circuit adapted to extract the signal amplitude or a power-law thereof.
 11. The system of claim 10, wherein the amplitude generator is comprised of a lowpass filter to attenuate generated harmonics and a power-law inverter.
 12. The system of claim 11, further comprising a linear quantizer adapted to convert the amplitude of the analytic signal into binary signals.
 13. The system of claim 10, wherein the amplitude generator is comprised of a nonlinear quantizer.
 14. The system of claim 9, wherein comparing an even power-law transform of the real and imaginary parts of the analytic signal to each other further comprises multiplying the even power-law transform of the real and imaginary parts of the analytic signal by multiple constant gains in their respective gain banks.
 15. The system of claim 14, further comprising a bank of comparators to generate a phase binary signal.
 16. The system of claim 9, further comprising respective logarithmic amplifiers which act on an even power-law transform of the real and imaginary parts of the analytic signal prior to their comparison to each other.
 17. A method converting a bandpass analog signal to a digital signal, comprising the steps of: generating real and imaginary parts of an analytic representation of said bandpass signal with a quadrature hybrid, wherein the real and imaginary parts of an analytic representation of said bandpass signal is an analytic signal; determining an amplitude of the analytic signal by adding an even power-law transform of the real and imaginary parts of the analytic signal; converting the signal amplitude into a digital amplitude signal with a quantizer; and converting a signal phase of the analytic signal into a digital phase signal by comparing the real and imaginary parts of the analytic signal to zero and comparing an even power-law transform of the real and imaginary parts of the analytic signal to each other.
 18. The method of claim 17, wherein the amplitude is lowpass filtered to attenuate generated harmonics previous to the quantizer
 19. The method of claim 17, wherein a power-law inverter transforms the amplitude prior to the quantizer. 